how to deconvolute a array ?
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i would like to deconvolute a matrix
%suppose that b is already the convolution of the array "a" with an array "c"
% I would like to deconvulte " b " to re-obtain "a" and "c".
% any idea how to do so?
% thanks you in advance
Chris Turnes on 9 Feb 2022
Deconvolution is equivalent to polynomial division. You can get the polynomial division and its remainder with the deconv function.
% Take two vectors.
a = randn(7,1);
c = randn(10,1);
% Compute their convolution.
b = conv(a, c);
% "Recover" the first by deconvolving c from b:
[ahat,r] = deconv(b, c);
% Check the residual and the remainder polynomial
However, it's important to note that this is not a least-squares solution to the deconvolution, and if b isn't really the result convolving something with c, you may not get an answer that's particularly close to the least squares result. To get the least squares result, you would construct the Toeplitz system corresponding to the convolution and solve it:
% Add some "noise" to c:
bhat = conv(a, c + 1e-3*randn(size(c)));
% Solve with deconv:
ahat_deconv = deconv(bhat, c);
% Compare convolving the result with c against the vector we started with:
norm(bhat - conv(ahat_deconv, c))
% Solve with least-squares:
T = convmtx(c, length(a));
ahat_ls = T \ bhat;
% Compare convolving the least-squares result with c against the vector we
% started with:
norm(bhat - conv(ahat_ls, c))
There are efficient algorithms to solve the Toeplitz system, though there are not any functions directly in MATLAB to do so.